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% Pin No.: 06293
% Paper ID: CDC02-INV2302
% Title: Warga derivate containers and other generalized differentials
% Paper Accepted Category: REGULAR INVITED
% Session Title: Geometric Methods in Control Theory, II: Special Sessions
% Dedicated to Jack Warga's 80th Birthday
% Session Code: FrP11-2
% Presentation Format: ORAL
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% The full Author's Kit is available for you at the CDC02 web site:
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%            This is a LateX file, containing  the paper 
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%   WARGA DERIVATE CONTAINERS AND OTHER GENERALIZED DIFFERENTIALS
%
%                       by H. J. Sussmann
%
%        This paper has been submitted for presentation at the 41th IEEE 
%        Conference on Decision and Control, to be held in Las Vegas,
%        December, 2002,  and  for publication in the conference  proceedings. 
%  
%               Date of this version: February 21, 2002
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\vspace{3cm}

{\Large  

\noindent This is the back of {\Huge Page \arabic{pagono}}\\
of the paper

\vspace{1cm}
\begin{center}
{\huge Warga derivate containers\\ 
and other generalized\\
differentials}

$\quad$\\

by\\

$\quad$\\ 
{\huge H\'ector J. Sussmann}\\

\end{center}
}

\newpage 

\null

\vspace{3.3cm}

\noindent Paper ID: $\,\,\,${\huge  CDC02-INV2302} \\

\noindent Session Title:\\

\medbreak
{\Large \noindent Geometric Methods in Control\\

\vspace{-5mm}\noindent Theory, II: Special Sessions\\

\vspace{-5mm}\noindent Dedicated to Jack Warga's\\

\vspace{-3mm}\noindent 80th Birthday}

\vspace{4mm}
Session Code: {\Huge FrP11-2} \\

Paper Category: REGULAR INVITED

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{\Large\em Dedicated to Jack Warga on his 80th birthday}\\
$\,$\\
H\'ector J.\ Sussmann\\
Department of Mathematics\\
Rutgers, the State University of New Jersey\\
Hill Center---Busch Campus\\
%110 Frelinghuysen Road\\
Piscataway, NJ 08854-8019, USA\\
{\small\tt sussmann@math.rutgers.edu}\\
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\begin{document}

\abovedisplayskip 4pt plus2pt minus5pt\belowdisplayskip \abovedisplayskip
\twocolumn[\titlestuff]
\footnotetext[1]{The  author was supported in part by
NSF Grant DMS98-03411-00798.}

\vskip4.5cm

\ssection{Introduction}
\label{sec-intro}

\noindent This is the first of two papers devoted to recent ideas
on the theory of generalized differentials with good open mapping properties.
Here we will discuss ``generalized differentiation 
theories'' (abbr. GDTs), 
with special 
\mbox{emphasis} on the series of developments initiated 
by Jack Warga's pioneering work on derivate containers.
In the second paper, we will focus on the most recent theory,
of ``path-integral generalized differentials,'' and prove that
it has the \mbox{crucial} properties required for a version of the 
Pontryagin \mbox{Maximum} \mbox{Principle} (\mbox{abbr.} PMP) 
to exist, namely, the chain rule and the directional open
\mbox{mapping} property. 

Our work continues the \mbox{study}
of general smooth, \mbox{nonsmooth}, high-order, and \mbox{hybrid} 
\mbox{versions} of the \mbox{PMP} for finite-dimensional
\mbox{deterministic}
\mbox{optimal} \mbox{control} \mbox{problems} without state \mbox{space}
\mbox{constraints}
by \mbox{means} of a method \mbox{developed} by us in recent years.
As \mbox{explained} in
\cite{sussshyb,susssmf2000,sussncn2000,susssyd2000}, 
such \mbox{versions} can be \mbox{derived} in a \mbox{unified} way, by using
a \mbox{modified} version of the \mbox{approach} of the classical book
\cite{pontrybook} by 
\mbox{Pontryagin} {\it et al.}. In the classical approach,
one
\mbox{constructs} ``\mbox{packets} of needle \mbox{variations},'' 
\mbox{linearly} \mbox{approximates} these
\mbox{packets} at the base \mbox{value} of the \mbox{variation} parameter,
and \mbox{propagates} the \mbox{resulting} \mbox{linear} approximations to the
terminal point of the \mbox{trajectory} by means of the \mbox{differentials}
of the \mbox{reference} flow maps.  This technique
must be \mbox{modified} by \mbox{(a)~replacing}
the the theory of the \mbox{classical} 
\mbox{differential} by
other GDTs, \mbox{(b)~replacing} the time-varying vector fields that
\mbox{occur} in the classical MP by {\em flows}, and (c)~replacing
the \mbox{needle} variations by {\em \mbox{abstract}
variations}.  Every GDT yields a \mbox{version} of the MP
provided it \mbox{satisfies}
some \mbox{natural} properties such as the chain rule
and an \mbox{appropriate} ``\mbox{directional} open 
\mbox{mapping} property'' (abbr. DOMP).
(\mbox{Details} are provided in
\cite{sussshyb,susssmf2000,sussncn2000,susssyd2000}.)

In these two papers we will not explicitly
discuss flows, variations, and the
PMP. How these topics are dealt with in the theory, and how
one gets versions of the PMP for general GDTs, has been extensively
disucssed in other papers, and for lack of space we choose not to address
these questions here. We will, instead, give a detailed
discussion of GDTs, and show how several important GDTs
are defined and how they are related to one \mbox{another}.
It will turn out that the complicated nature of these interrelationships,
and in particular the fact that none of the ``natural'' theories
contains all the others, suggests the question whether
a theory that truly contains all other GDTs can be defined.
The answer to this question is given in the second paper of the series,
where a theory that does the desired job is defined and studied.

Here we propose an axiomatic definition of the \mbox{concept} of a GDT
and a precise statement of the 
\mbox{DOMP}. We then outline the definitions of
some \mbox{GDTs}, such as the Warga derivate containers (abbr. 
\mbox{WDCs}), and our more recent theories of 
\mbox{``multidifferentials''} and  ``\mbox{generalized}
differential quotients'' (abbbr. GDQs).  Multidifferentials are
generalizations of Warga's derivate containers, which in turn include
the Clarke \mbox{generalized} Jacobians (abbr. CGJs) as a special case. The
GDQs, on the other hand, generalize the CGJs in a \mbox{different}
\mbox{direction}, and constitute a 
\mbox{theory} that \mbox{neither} 
\mbox{contains} nor is contained in that
of the WDCs or in that of \mbox{multidifferentials.} We will make this
precise by \mbox{comparing} these two types of theories by means
of \mbox{examples}. This will set the stage for
the second \mbox{paper}, in which we will presents a theory that achieves the
\mbox{desired} unification.

\begin{remark}
Generalized differentials that extend the classical differential
and the Warga derivate containers were studied by H. Halkin
in the 1970s in three remarkably insightful papers 
(\cite{halkina, halkinb, halkinc}) that, unfortunately, 
do not seem to have attracted the attention they deserved.

The work presented here is a continuation and extension of 
that of Halkin. In particular, our machinery makes it possible to deal 
systematically with set-valued maps, which appear when one studies
flows of continuous but not necessarily Lipschitz vector fields
(as in the ``\L ojasiewicz maximum principle,'' cf. 
\cite{susspavel}) 
and also in the analysis of differential inclusions (cf.
\cite{susskobe}). \proofend
\end{remark}





\ssection{GDTs}\label{gdts}

\noindent A ``generalized differentiation theory'' (abbr. GDT) is, roughly,
a way of assigning a ``differential'' $\mathcal{D}(F;x,y;S)$ 
to each 4-tuple $(F,x,y,S)$ consisiting
of (a)~a set valued map $F:M\tto N$ whose source $M$ and target 
$N$ are manifolds of class $C^1$,
(b)~a point $x$ of $M$, (c)~a point $y$ of $N$, and (d)~a subset
$S$ of $M$.  The object $\mathcal{D}(F;x,y;S)$ is required
to be a set of nonempty compact subsets of ${\rm Lin}(T_xM,T_yN)$, the
space of linear maps from $T_xM$ to $T_yN$, where, for a manifold
$Q$ and a $q\in Q$, $T_qQ$ denotes
the tangent space to $Q$ at $q$.
The members of $\mathcal{D}(F;x,y;S)$ are the 
``$\mathcal{D}$-differentials of $F$ at $(x,y)$ in the direction
of $S$.''  (The  set  $\mathcal{D}(F;x,y;S)$ could be empty, in which
case we say that $F$ is not 
$\mathcal{}D$-differentiable at $(x,y)$ in the direction
of $S$.)  The correspondence $\mathcal{D}$ is required to satisfy the
chain rule, the Cartesian product rule, locality, and invariance under
$C^1$ diffeomorphisms.  Furthermore, it is required to be an extension
of the classical theory of differentials of maps of class $C^1$.

To make all this precise, we introduce some notations and definitions.

\skipopag

\noindent{\em Set-valued maps.}
By a  {\em set-valued map} (abbr. SVM) we mean a triple 
$F=(A,B,G)$ such that $A$ and $B$ are 
\mbox{sets} and $G$ is a subset of
$A\times B$. The \mbox{sets} $A$, $B$, $G$ are, 
\mbox{respectively}, the {\em
source}, {\em \mbox{target}}, and {\em graph} of $F$, and we
write $A=\so(F)$, 
\mbox{$B=\ta(F)$}, \mbox{$G=\gr(F)$}.  If $x$ is any object, we
write \mbox{$F(x)=\{y:(x,y)\in\gr(F)\}$}.  (Hence $F(x)=\emptyset$ unless
\mbox{$x\in\so(F)$}.)  The \mbox{sets}
\mbox{$\dom(F)=\{x\in\so(F):F(x)\neq\emptyset\}$},
$\im(F)=\bigcup_{x\in\so(F)}F(x)$, are, respectively, the {\bem
domain} and {\em image} of $F$. If $F=(A,B,G)$ is an SVM, we
say that $F$ is an {\em SVM from $A$ to $B$}, and write
$F:A\tto B$.  We use $SVM(A,B)$ to denote the set of all SVMs
from $A$ to $B$.
The expression ``ppd map'' stands for ``\mbox{possibly}
\mbox{partially} defined
(that is, not necessarily everywhere \mbox{defined})
ordinary (that is, single-valued) map,'' and we write
$f:A\brokarrow B$ to indicate that $f$ is a ppd map from $A$ to
$B$.  A {\it time-varying ppd map} from a set $A$ to a set $B$ is a
ppd map from $A\times\rr$ to $B$.

\medbreak\noindent{\em Classes of manifolds.}
We use $\zz_+$ to denote the set of all nonnegative
\mbox{integers}. If $k\in\zz_+$, then $\mathcal{M}^k$,
$\mathcal{SVM(M}^k)$ will denote, \mbox{respectively}, the class of
\mbox{all}
finite-dimensional Hausdorff manifolds of class $C^k$ \mbox{without}
boundary, and the class of all \mbox{SVMs} $F$ such that
$\so(F)\!\in\!\mathcal{M}^k$ and $\ta(F)\!\in\!\mathcal{M}^k$.  If
$k\ge 1$ and \mbox{$x\!\!\in\!\! M\!\!\in\!\!\mathcal{M}^k$}, then
$T_xM$, $T_x^*M$, $TM$, $T^*M$ will denote, respectively, the
\mbox{tangent} and cotangent \mbox{spaces} to $M$ at $x$, and the
tangent and cotangent bundles of $M$. Clearly, $TM$ and $T^*M$ belong
to $\mathcal{SVM(M}^{k-1})$.

\medbreak\noindent{\em The Bouligand tangent cone.}
If $M\in\mathcal{M}^1$, $x\in M$, and $S\subseteq M$, then
$T_x^BS$ will denote the {\em Bouligand \mbox{tangent} cone} to $S$ 
at $x$. By \mbox{definition}, $T_x^BS$  is the set of
\mbox{all} tangent vectors $v\in T_xM$ such that there exist a 
\mbox{sequence}
$\{(x_j,h_j)\}_{j\in\nn}$ of points of $S\times\rr$ having the property that
$h_j>0$ for all $j$, $h_j\downarrow 0$ as $j\to\infty$, and 
$\lim_{j\to\infty}h_j^{-1}(\varphi(x_j)-\varphi(x))
=v\varphi$ for all functions $\varphi\in C^1(M,\rr)$.

\medbreak\noindent{\em Classes of linear spaces, linear 
multimaps.}
We write $\mathcal{RLS}$,  $\mathcal{FDRLS}$, 
to denote, respectively,  the class of all linear spaces over $\rr$, and 
the class of all $X\in\mathcal{RLS}$ that are finite-dimensional.
If $X,Y\in\mathcal{RLS}$,  then 
${\rm Lin}(X,Y)$ denotes the set of all $\rr$-linear maps from
$X$ to $Y$. A \mbox{subset} of ${\rm Lin}(X,Y)$ will be called a {\em 
\mbox{linear} \mbox{multimap}} from $X$ to $Y$. 
We use ${\rm MLin}(X,Y)$, ${\rm MLin}_c(X,Y)$,
to denote, \mbox{orespectively}, 
the set of all linear  multimaps from $X$ to $Y$
and (if $X,Y\in\mathcal{FDRLS}$) the set of \mbox{all} nonempty 
compact linear  multimaps from $X$ to $Y$.

\begin{definition}\label{defpfgdt}
A {\em generalized differentiation theory} (abbr. GDT)
is a correspondence $\mathcal{D}$ that \mbox{assigns} to 
\mbox{every} $F\in\mathcal{SVM}(\mathcal{M}^1)$,
every \mbox{$(x,y)\in\so(F)\times\ta(F)$},
and \mbox{every} subset $S$ of the source $\so(F)$, 
a set \mbox{$\mathcal{D}(F;x,y;S)\subseteq {\rm MLin}_c(T_xM,T_yY)$}
of nonempty 
\mbox{compact} linear multimaps from $T_xM$ to $T_yY$,
in such a way that the following axioms are satisfied:

\ben 
\vspace{-6pt}
\item[1.] If $m,n\in\zz_+$, $f\!:\!\rr^m\!\mapsto\!\rr^n$ is a map 
of class $C^1$, and $S$ is a polyhedral convex cone in $\rr^m$, then
$\{Df(0)\}\!\in\!\mathcal{D}(f;0,0;S)$. 

\comment{
\vspace{-8pt}
\item[2.] ({\bem Monotonicity}) Whenever 
$F_1,F_2\in\mathcal{SVM}(\mathcal{M}^1)$,
$M=\so(F_1)=\so(F_2)$, $N=\ta(F_1)=\ta(F_2)$, $x\in M$, $y\in N$, 
$S\subseteq M$, $\Lambda\in\mathcal{D}(F_1;x,y;S)$, 
$\Lambda'\in{\rm MLin}_c(T_xM,T_yN)$, 
$\gr(F_1)\subseteq\gr(F_2)$ and 
$\Lambda\subseteq\Lambda'$, it follows that
$\Lambda'\in\mathcal{D}(F_2;x,y;S)$.
}

\vspace{-8pt}
\item[2.] {\bem (The chain rule)}  If $F_i\!\in\!\mathcal{SVM}(\mathcal{M}^1)$
and \mbox{$S_i\!\subseteq\! M_i$} for $i\!=\!1,2$,  
$M_1\!=\!\so(F_1)$, \mbox{$M_2\!=\!\ta(F_1)\!=\!\so(F_2)$}, 
\mbox{$M_3\!=\!\ta(F_2)$}, $x_i\!\in\! M_i$ for $i\!=\!1,2,3$,
$F_1(S_1)\!\subseteq\! S_2$, $\Lambda_1(T_{x_1}^BS_1)\!\subseteq\!
T_{x_2}^BS_2$, and
$\Lambda_2\!\in\!\mathcal{D}(F_2;x_{2},x_{3};S_2)$, then 
\mbox{$\Lambda_2\circ\Lambda_1\!\in\!\mathcal{D}(F_2\circ F_1;x_{1},x_3;S_1)$}.

\comment{
if either (a)~$F_1$ is a diffeomorphism of class
$C^1$ from a neighborhood $U_1$ of $x_1$ onto a neighborhood
$U_2$ of $x_2$, $F_1(x_1)=x_2$, $F_1(S_1\cap U_1)=S_2\cap U_2$, 
and \mbox{$\Lambda_1=\{DF_1(x_1)\}$}, or
(b)~$F_1$ is a single-valued map of class $C^1$ \mbox{defined} on a 
neighborhood $U$ of $x_1$, \mbox{$F_1(x_1)=x_2$},
\mbox{$\Lambda_1\in\mathcal{D}(F_1;x_{1},x_{2};S_1)$},
and \mbox{$\Lambda_1=\{DF_1(x_1)\}$}, or
(c)~$\Lambda_1\in\mathcal{D}(F_1;x_{1},x_{2};S_1)$,
$\Lambda_1\cdot T_{x_1}^BS_1\subseteq T_{x_2}^BS_2$ and,
for some coordinate chart $\kappa\!:\!U\!\to\!\rr^{n}$ on $M_2$
such that \mbox{$x_2\!\in\! U$} and $\kappa(x_2)=0$, there is
a \mbox{polyhedral} cone $C$ in $\rr^{n}$ such that 
\mbox{$\kappa(S_2\cap U)=C\cap\kappa(U)$}.}

\vspace{-8pt}
\item[3.] {\bem (The product rule)}  If, for $i=1,2$,  
$F_i\in\mathcal{SVM}(\mathcal{M}^1)$, 
$x_i\!\in\! M_ i\!=\!\so(F_i)$, \mbox{$y_i\!\in\! N_ i\!=\!\ta(F_i)$}, 
$S_i\subseteq M_i$, 
and $\Lambda_i\in\mathcal{D}(F_i;x_{i},y_i;S_i)$, then 
$\Lambda_1\times\Lambda_2$ belongs to 
$\mathcal{D}(F_1\times F_2;(x_1,x_2),(y_1,y_2);S_1\times S_2)$.

\vspace{-8pt}
\item[4.] {\bem ($C^1$ invariance)}  Assume that
(a)~for $i=1,2$,  $M_i,N_i\in\mathcal{M}^1$, 
$F_i\in\mathcal{SVM}(\mathcal{M}^1)$, 
$M_i\!=\!\so(F_i)$, \mbox{$N_i\!=\!\ta(F_i)$}, 
$S_i\subseteq M_i$, $x_i\in M_1$, $y_i\in N_i$, 
$U_i,V_i$ are open
subsets of $M_i,N_i$ such that $x_i\in U_i$ and  $y_i\in V_i$,
(b)~$\Phi:U_1\mapsto U_2$ and $\Psi:V_1\mapsto V_2$ are 
diffeomorphisms of class $C^1$ such that 
$\Phi(x_1)\!=\!x_2$ and $\Psi(y_1)\!=\!y_2$, 
(c)~$\Phi(S_1\!\cap\! U_1)\!=\!S_2\!\cap\! V_2$, and
(d)~the set \mbox{$(\Phi\!\times\!\Psi)\Big(\gr(F_1)\!\cap\!((S_1\cap U_1)
\!\times\!V_1)\Big)$} is equal to
$\gr(F_2)\!\cap\!((S_2\cap U_2)\!\times\! V_2)$. 
Then the set \mbox{$\mathcal{D}(F_2;x_2,y_2;S_2)$} is equal  to
\mbox{$D\Psi(y_1)\circ\mathcal{D}(F_1;x_1,y_1;S_1)
\circ D\Phi^{-1}(x_2)$}. \proofend
\een
\end{definition}

\begin{remark} 
The $C^1$ invariance property implies in particular that a GDT 
$\mathcal{D}$ is {\em local\/}, in the sense that
$\mathcal{D}$ satisfies

\vspace{-8pt}
\ben
\item[4'.] {\bem (Locality)}  Assume that (a)~$M,N\in\mathcal{M}^1$, 
\mbox{$x\in M$}, \mbox{$y\in N$}, $U,V$ is are open
subsets of $M,N$ such that $x\in U$, $y\in V$,
(b)~for $i=1,2$,  \mbox{$F_i\in\mathcal{SVM}(\mathcal{M}^1)$}, 
$M\!=\!\so(F_i)$, \mbox{$N\!=\!\ta(F_i)$}, 
$S_i\subseteq M_i$, \mbox{(c)~$S_1\cap U=S_2\cap U$}
and \mbox{$\gr(F_1)\cap((S_1\cap U)\times V)$}
is equal to \mbox{$\gr(F_2)\cap((S_2\cap U)\times V)$}. Then
$\mathcal{D}(F_1;x,y;S_1)=\mathcal{D}(F_2;x,y;S_2)$.\proofend
\een
\end{remark}

\begin{remark} 
To construct a GDT $\mathcal{D}$, it suffices to \mbox{specify}
$\mathcal{D}(F;0,0;S)$, for all $k,\ell$, 
whenever $F:\rr^k\tto\rr^\ell$
and $S\subseteq\rr^k$. Let $\mathcal{D}_0$ be a correspondence
of the kind described above, but restricted to
$M=\rr^k$, $N=\rr^k$, $x=0$, $y=0$. It is then a routine
matter to write down a list of axioms that will guarantee
that $\mathcal{D}_0$ ``gives rise in a canonical way''
to a GDT in the sense of
Definition \ref{defpfgdt}. 

The crucial point is $C^1$ invariance and locality, i.e., the version
of the $C^1$ invariance condition of Definition \ref{defpfgdt}
restricted to $\rr^k$, $\rr^\ell$. We omit the details of the
general theory, but refer the reader to our discussion of
weak multidifferentials below (in \S \ref{seconwmd}, especially
the remarks following Corollary  \ref{chainrulecor}), where 
we explain how this is done in one particular case. \proofend
\end{remark}

\ssection{The directional open mapping property}\label{domptrans}

\noindent We say that a GDT $\mathcal{D}$ has the {\bem directional 
open mapping property} if the following statement is true:.

\medbreak
\noindent{\bf (DOMP)}\ {\em 
Assume that $n,m\in\zz_+$, 
$F\!\in\!\mathcal{SVM}(\rr^n,\rr^m)$,  $v\!\in\!\rr^m$,
\mbox{$C$ is a closed} convex cone in $\rr^n$, 
$\Lambda$ \mbox{belongs} to $\mathcal{D}(F;0,0;C)$, and 
\mbox{$v\!\in\!\bigcap_{L\in\Lambda}{\rm Int}(LC)$}. Then 
there exists a closed convex cone $D$ in $\rr^n$ such that
$v\in{\rm Int}(D)$, having the property that
for every $\delta\!\in]0,\infty[$ there exists
an $\eps(\delta)\!\in]0,\infty[$ such that
\fbox{$D\!\cap\!\{y\!\in\!\rr^m\!:\!\|y\|\!\le\!\eps(\delta)\}\!\subseteq\!
F(C\!\cap\!\{x\!\in\!\rr^n\!:\!\|x\|\!\le\!\delta\})$}.}\proofend

\ssection{Warga derivate containers (WDCs)}\label{sec-wdc}

\vspace{-2mm}
\begin{definition}\label{tomimic}
Let $f$ be a map from an open subset $\Omega$ of $\rr^m$ to $\rr^n$, and let $x_*\in\Omega$.
A {\em Warga derivate container} of $f$ at $x$ is a nonempty compact 
subset $\Lambda$ of ${\rm Lin}(\rr^m,\rr^n)$ such that 
\ben
\item[(*)] for every open neighborhood $\Lambda'$ of $\Lambda$ 
in the space ${\rm Lin}(\rr^m,\rr^n)$ 
\ there \ exist \ an \ open \ neighborhood
\een 
\skipopag
\ben
\item[]  \mbox{$U=U_{\Lambda'}$ of $x_*$ in $\Omega$} and a sequence
$\{f_j\}_{j=1}^\infty$ of maps of
class $C^1$ from $U$ to  $\rr^n$, such that 
(a)~$f_j\to f$ uniformly on $U$ as $j\to\infty$, and 
(b)~the differential $Df_j(x)$ belongs to $\Lambda'$ for 
every $x\in U$ and every $j$.
\een
\end{definition}

It follows from the definition of a WDC that any map $f$ that admits
a WDC at a point $x_*$ must be Lipschitz continuous on a neighborhood
of $x_*$, since we can \mbox{always} choose $\Lambda'$ to be bounded,
in which case $f$ will be a \mbox{uniform} limit, on a neighborhood
$U$ of $x_*$, of maps of class $C^1$ with uniformly bounded
derivatives.

Associated to the Warga derivate containers we can define  GDTs
by extending the definition of the WDCs to ``WDCs along sets.''
This can be done in more than one way. For example, 
suppose $C$ is a closed convex cone 
in $\rr^n$,  $U$ is a neighborhood of $0$ in $\rr^n$, 
and $f:U\cap C\mapsto\rr^m$ is a Lipschitz continuous
map. We could then define a WDC of $f$ at $0$ in the direction
of $C$ to be a WDC of any extension $\tilde f$ of 
$f$ to a Lipschitz map defined on  a full neighborhood
$V$ of $0$ in $\rr^n$. Alternatively, 
we could mimic Definition \ref{tomimic}, using uniform
approximations of $f$ by maps of class $C^1$
on relative neighborhoods  $V\cap C$ of $0$ in $C$. These
two approaches lead to different theories.

We will not pursue this matter here any further
and will, instead, move on directly
to a more general theory called WMD (``weak multidifferentials'')
that will be defined {\it ab initio} as theory of differentials 
in the direction of sets. 

\vbox{
\ssection{Regular set-valued maps}\label{sec-regsetval}

\noindent If $X$, $Y$ are metric spaces, then 
$SVM_{comp}(X,Y)$ will denote the subset of 
$SVM(X,Y)$ whose \mbox{members} are
the set-valued maps from $X$ to $Y$
that have a \mbox{compact} graph.
We say that a sequence  $\{F_j\}_{j\in\nn}$ of 
\mbox{members} of $SVM_{comp}(X,Y)$ 
{\bem inward graph-converges} to an $F\in SVM_{comp}(X,Y)$---and
write $F_j\togr F$---if
for every open subset $\Omega$ of $X\times Y$ such that 
$\gr(F)\subseteq\Omega$ there exists a
$j_\Omega\in\nn$ such that $\gr(F_j)\subseteq\Omega$
whenever $j\ge j_\Omega$.
}


\begin{definition}\label{defregu} 
Assume that $X$, $Y$ are  metric spaces.  
A {\bem regular set-valued map} from $X$ to $Y$ is a set-valued map 
$F\in{}SVM(X,Y)$ such that 
\begin{itemize}
\vspace{-6pt}
\item{} for every compact subset $K$ of $X$, the restriction
$F\restr K$ of $F$ to $K$ belongs to 
$SVM_{comp}(K,Y)$  and is a limit---in the sense of inward 
graph-convergence---of a sequence of continuous
single-valued maps from $K$ to $Y$. 
\end{itemize}
\vspace{-6pt}\noindent We use $\regu{X}{Y}$ to denote the set of all regular
\mbox{set-valued}  maps from $X$ to $Y$. \proofend
\end{definition}
It is easy to see that if $F:X\mapsto Y$ is an ordinary 
(that is, single-valued and everywhere defined) map, then
$F$ belongs to $\regu{X}{Y}$ if and only if $F$ is continuous.

An important class of examples of regular maps is \mbox{provided} by the
following two results, whose proof we omit.
\begin{theorem}\label{kakucon}
Assume that $K$ is a compact metric 
\mbox{space},  $Y$ is a normed 
space, and  $C$ is a convex subset of $Y$. Let $\Phi:K\tto C$ be a 
set-valued map such that the graph of $\Phi$ is compact and the value 
$\Phi(x)$ is a nonempty convex set for every $x\in K$. Then $\Phi$ is regular
as a map from $K$ to $C$.
\end{theorem}
\begin{theorem}\label{kakucon-nocomp}
Assume that $X$ is a  metric space, $Y$ is a normed 
space, and $C$ is a convex subset of $Y$. Let $\Phi:K\tto C$ be an 
upper semicontinuous set-valued map with nonempty compact
convex values.  Then $\Phi\in\regu{X}{C}$.\proofend
\end{theorem}

In addition, it is not hard to prove the following.
\begin{theorem}\label{comporeg-13}
Assume that $X$, $Y$, $Z$ are metric \mbox{spaces}. Let
\mbox{$F\in\regu{X}{Y}$}, $G\in\regu{Y}{Z}$. Then the
\mbox{composite} map
$G\circ F$ belongs to $\regu{X}{Z}$.\proofend
\end{theorem}

\ssection{Weak multidifferentials}\label{seconwmd}

\noindent ``Multidifferentials'' were studied in \cite{sussmul}. Here we 
\mbox{introduce} a slightly more general theory, of objects that we call
``weak multidifferentials.''

As a preliminary, we need some notations.
If $X$, $Y$ are finite-dimensional real spaces,
$\Omega\subseteq X$, and $\Lambda\subseteq{\rm Lin}(X,Y)$, we use 
$C_{\Lambda}^1(\Omega,Y)$ to denote the set of all maps
$h:\Omega\mapsto Y$ of class $C^1$ such that 
$Dh(x)\in\Lambda$ for all $x\in\Omega$. (The precise meaning
of ``$h$ is of class $C^1$'' is clear if $\Omega$ is open.
We also need to assign a meaning to this expression when
$\Omega$ is closed. This is done in the usual Whitney way,
which turns out to be equivalent to the following: a map
of class $C^1$ from $\Omega$ to $Y$ is a pair $(h_1,H)$ 
consisting of continuous maps \mbox{$h_1:\Omega\mapsto Y$}, 
$H:\Omega\mapsto{\rm Lin}(X,Y)$, having the property
that, for every $x\in\Omega$, 
$h_1(x')-h_1(x)-H(x)\cdot(x'-x)=o(\|x'-x\|)$ as $x'\to x$
via values in $\Omega$. 
If $h=(h_1,H)$ is such an object,
then $Dh$ is defined to be $H$, and if $x\in\Omega$ we define
$h(x)$ to be $h_1(x)$.)

\begin{definition}
Let $X$, $Y$ be finite-dimensional normed real 
\mbox{linear} spaces. Define
$\bar B_X=\{x\in X:\|x\|\le 1\}$,  $Z={\rm Lin}(X,Y)$. Let 
$F:X\tto Y$ be a set-valued map. Let $(x_*,y_*)\in X\times Y$,
and let $\Lambda$ be a compact subset of $Z$. Let $C$ be a 
closed convex cone in
$X$.  We say that $\Lambda$ is a {\sl weak multidifferential of $F$ at
$(x_*,y_*)$ in the direction of $C$}, and write
$$
\Lambda\in WMD(F;x_*,y_*;C)\,,
$$
if the following is true:

\medbreak\noindent(WMD) for every neighborhood $\Lambda'$ of 
$\Lambda$ in $Z$ there exists a pair $(R,\Theta)$ such that
\bi\item[]\bi\bi
\item[(WMD.1)] $R>0$, and $\Theta$ is a function on the interval
$]0,R]$ with values in $[0,\infty[$ and such that 
$\lim_{s\downarrow 0}\Theta(s)=0$, 
\item[(WMD.2)] for every $r\in]0,R]$ there exist  $f,h$  such that
\bi
\item[(WMD.2.1)] $f\in{\rm REG}(x_*+(C\cap r\bar{B}_X),Y)$;
\item[(WMD.2.2)] $\gr(f)\subseteq\gr(F)$;
\item[(WMD.2.3)] $h\in C^1_{\Lambda'}(x_*+(C\cap r\bar{B}_X),Y)$;
\item[(WMD.2.4)]  $h(x_*)=y_*$;
\item[(WMD.2.5)] the inequality  
\ei\ei\ei\ei
\beq\label{issatweak}
\sup\Big\{|\!|y\!-\!h(x)|\!|:y\!\in\!f(x)\Big\}\!\le\!r.\Theta(r)
\eeq 
\bi\item[]\bi\item[]\bi\item[]\bi
holds whenever   $x-x_*\in C\cap r \bar{B}_X$. \mybox
\ei\ei\ei\ei
\end{definition}

Weak multidifferntials satisfy the following  version of the chain rule:

\skipopag

\begin{theorem}\label{chainrule}
Let $X_1,X_2,X_3$ be finite-dimensional \mbox{real} linear spaces, and
let $\bar x_i\in X_i$, $i=1,2,3$. For \mbox{$i=1,2$}, let 
$F_i:X_i\tto X_{i+1}$ be a set-valued map from $X_i$ to $X_{i+1}$, 
let $C_i$ be a closed convex cone in $X_i$, and let
$\Lambda_i$ be %a compact subset of $L(X_i,X_{i+1})$
such that 
$\Lambda_i\in WMD(F_i;\bar x_i,\bar x_{i+1},C_i)$. Let
$S$ be the linear subspace of $X_2$ spanned by $C_2$, and let
$\Pi$ be a linear projection from $X_2$ onto $S$.  Assume that
$F_1(\bar x_1+C_1)$ is a subset of $\bar x_2+C_2$.
Then 
\beq
\Lambda_2\compo\Pi\compo\Lambda_1\in WMD(F_2\compo F_1;\bar x_1,
\bar x_3;C_1)\,.
\eeq
\end{theorem}
Theorem \ref{chainrule} was proved in  \cite{sussmul} for multidifferentials,
but the same proof applies to weak multidifferentials as well and we will
not repeat it here. 

The following is a trivial consequence of Theorem \ref{chainrule}.
\begin{corollary}\label{chainrulecor}
Let $X_1,X_2,X_3$ be finite-dimensional \mbox{real} linear spaces, and
let $\bar x_i\in X_i$, $i=1,2,3$. For \mbox{$i=1,2$}, let 
$F_i:X_i\tto X_{i+1}$ be a set-valued map from $X_i$ to $X_{i+1}$, 
let $C_i$ be a closed convex cone in $X_i$, and let
$\Lambda_i$ be %a compact subset of $L(X_i,X_{i+1})$
such that 
$\Lambda_i\in WMD(F_i;\bar x_i,\bar x_{i+1},C_i)$. Let
$S$ be the linear subspace of $X_2$ spanned by $C_2$, and let
$\Pi$ be a linear projection from $X_2$ onto $S$.  Assume that
$F_1(\bar x_1+C_1)$ is a subset of $\bar x_2+C_2$ and
$LC_1\subseteq S$ for all $L\in\Lambda_1$. Then 
\beq
\Lambda_2\compo\Lambda_1\in WMD(F_2\compo F_1;\bar x_1,
\bar x_3;C_1)\,.
\eeq
\end{corollary}
Using Corollary \ref{chainrulecor}, we can extend WMD to a
GDT by defining $WMD(F;x_*,y_*;S)$, if $M$, $N$ are 
\mbox{manifolds} of class $C^1$, $F:M\tto N$, $x_*\in M$,
$y_*\in N$, and $S\subseteq M$, as follows. We call the set
$S$ {\em conic at $x_*$} if there \mbox{exists} a coordinate chart
$\kappa$ near $x_*$ such that 
$\kappa(x_*)=0$ and $\kappa(\dom(\kappa)\cap S)=
\im(\kappa)\cap C$ for some closed convex cone 
$C$ in $\rr^{{\rm dim}\,M}$.  If $S$ is conic at $x_*$, then
we \mbox{define} $WMD(F;x_*,y_*;S)$  as a set of nonempty compact
subsets of ${\rm Lin}(T_{x_*}M,T_{y_*}N)$  using a chart $\kappa$ as above,
\mbox{together} with a chart for $N$ near $y_*$.
\mbox{Corollary} \ref{chainrulecor} 
then tells us that $WMD(F;x_*,y_*;S)$  does not depend
on the choice of charts, so it is intrinsically defined.
We complete the definition by taking $WMD(F;x_*,y_*;S)$
to be empty if $S$ is not conic at $x_*$. 

It is then easy to show that 
\begin{theorem}
WMD is a GDT that has the DOMP.\proofend
\end{theorem}

\medbreak
\ssection{Generalized differential quotients (GDQs)}\label{sec-gdq}

\begin{definition}\label{defquasi} 
Let $m,n\in\zz_+$, let $F:\rr^m\tto\rr^n$ be a set-valued
map, and let $\Lambda$ be a nonempty compact subset of $\rr^{n\times m}$. 
Let $S$ be a subset of  $\rr^m$.  
We say that $\Lambda$ is a {\bem generalized differential quotient}
(abbreviated ``GDQ'') {\bem of 
$F$ at $(0,0)$ in the direction of $S$}, and write
$\Lambda\in  GDQ(F;0,0;S)$, 
if for every positive real number $\delta$ there exist $U$, $G$ such that 
\ben
\vspace{-6pt}
\item[1.] $U$  is a  compact neighborhood of $0$ in $\rr^m$ and $U\cap S$
is compact;

\vspace{-8pt}
\item[2.] $G$  is a  regular set-valued map from $U\cap S$ to the 
$\delta$-neighborhood  $\Lambda^\delta$ of $\Lambda$ in $\rr^{n\times m}$;

\vspace{-8pt}
\item[3.] $G(x)\cdot x\subseteq F(x)$
for every $x\in U\cap S$.\proofend
\een
\end{definition}

\begin{remark}
The definition of the GDQ is a generalization of a characterization
of classical differentials due to Botsko and Gosser, \cite{botsko}.\proofend
\end{remark}



If $M$, $N$ are $C^1$ manifolds, 
\mbox{$\bar x\!\in\! M$, $\bar y\!\in\! N$, $S\!\subseteq \!M$}, 
and \mbox{$F:M\tto N$}, then we can define a set 
$GDQ(F;\bar x,\bar y;S)$ of nonempty compact linear \mbox{multimaps}
from $T_{\bar x}M$ to $T_{\bar y}N$ 
by picking coordinate charts \mbox{$M\ni x\to\xi(x)\in\rr^m$}, 
\mbox{$N\ni y\to\eta(y)\in\rr^n$}---where \mbox{$m={\rm dim}\,M$,
$n={\rm dim}\,N$}---defined near $\bar x$, $\bar y$ such n 
that $\xi(\bar x)=0$, $\eta(\bar y)=0$, and
declaring a subset $\Lambda$ of ${\rm Lin}(T_{\bar x}M,T_{\bar y}N)$
to \mbox{belong} to $GDQ(F;\bar x,\bar y;S)$  if 
the multimap 
$D\eta(\bar y)\circ\Lambda\circ D\xi(\bar x)^{-1}$ is in $GDQ(\eta\circ
F\circ\xi^{-1};0,0;\xi(S))$.  It turns out that, with this definition,
{\bem the set $GDQ(F;\bar x,\bar y;S)$  does not depend on the choice 
of the charts $\xi$, $\eta$.} Moreover, the following three results
can be proved.

\begin{theorem}\label{isgdt}
GDQ is a generalized differentiation \mbox{theory}
that has the DOMP.\proofend
\end{theorem}

\begin{theorem}\label{contidif}
If $F:\rr^n\mapsto\rr^m$ is a continuous map,
$x\in\rr^n$, and $F$ is classically differentiable at $x$, then
$\{DF(x)\}\in GDQ(F;x,F(x);\rr^n)$.\proofend
\end{theorem}

\begin{theorem}\label{conticlark}
If $F:\rr^n\mapsto\rr^m$ is Lipschitz-continuous,
and $x\in\rr^n$, then the Clarke generalized Jacobian
$\partial F(x)$ belongs to $GDQ(F;x,F(x);\rr^n)$.\proofend
\end{theorem}

\ssection{Comparison of WDC, WMD, and GDQ}\label{sec-compar}

\begin{theorem}\label{contiwmd}
If $F:\rr^n\mapsto\rr^m$ is Lipschitz-continuous,
$x_*\in\rr^n$, and $\Lambda$ is a Warga derivate
container of $F$ at $x_*$, then $\Lambda\in WMD(F\;x_*,F(x_*):\rr^n)$.
\end{theorem}

\noindent {\em Proof.}\ Let $\Lambda'$ be a neighoborhood
of $\Lambda$ in $\rr^{m\times n}$. Let $\{F_j\}_{j=1}^\infty$
be a sequence of $\rr^m$-valued maps of class $C^1$ defined
on a closed ball $B$ in $\rr^n$ with center $x_*$, such that
$F_j\to F$ uniformly on $B$ and $DF_j(x)\in\Lambda'$ for all
$j$ and all $x\in B$. Pick $R$ and $\Theta$ in an arbitrary fashion
so that (WMD.1) holds and $\Theta(s)>0$ when $s>0$. Given $r\in]0,R]$, let 
$f$ be the restriction of $F$ to $B$. Then 
$f$ is a single-valued continuous map, so $f$ is {\it a fortiori}
a regular set-valued map. 

The maps \mbox{$B\ni x\mapsto F_j(x)-F_j(x_*)+F(x_*)=h_j(x)$}
converge uniformly to $f$
on $B$ and satisfy \mbox{$h_j(x_*)=F(x_*)$}. Pick $j$ such that
$\|h_j\|_{sup}\le r.\Theta(r)$, and let $h=h_j$, $C=\rr^n$. 
It is  then clear that all the conditions of (WMD.2)
are satisfied. \proofend

\medbreak Theorems \ref{contidif} and \ref{conticlark}
show that GDQ is a common generalization of both 
the classical theory of differentials and the CGJ. 
Furthermore, it is easy to exhibit maps that have  GDQs at a point $\bar x$
but are not classically 
\mbox{differentiable} at $\bar x$  and do not have 
a CGJ or a WDC. (A simple 
example is provided by the function $f:\rr\to\rr$ 
\mbox{given} by 
$f(x)=x\sin{1/x}$ if $x\neq 0$, $f(0)=0$.  Then
$f$ is not differentiable at $0$ and does not have
a WDC of $f$ at $0$, since $f$ is not Lipschitz continuous 
near $0$. On the other hand, the interval
$[-1,1]$ clearly belongs to $GDQ(f;0,0,\rr)$.)


The previous remarks show that GDQ is ``much more general
than the classical differential and the CGJ combined.''
It would then be natural to conjecture that
GDQ is also more general than WMD, or at least
more general than WDC. This, however, it not true,
because {\em there are Warga derivate containers
that are not GDQs.} 

For a \mbox{simple} example, consider
the function \mbox{$f:\rr\mapsto\rr^2\sim\cc$} \mbox{given} by
$f(x)=\int_0^xe^{\frac{i}{t}}\,dt$. Then $f$ is Lipschitz 
\mbox{continuous}.
Let \mbox{$h_j(t)=t^{-1}$} when 
\mbox{$|t|\ge j^{-1}$}, and $h_j(t)=j^2t$ when \mbox{$|t|\le j^{-1}$}.
Let \mbox{$f_j(x)=\int_o^xe^{i\,h_j(t)}\,dt$}. Then 
\mbox{$f_j\in C^1(\rr,\cc)$},
and $f_j\to f$ uniformly on compact \mbox{sets} as $j\to\infty$.
Furthermore, for 

\skipopag

\noindent every $x$ and \mbox{every} $j$, 
the derivative $f'_j(x)$ belongs to the unit
\mbox{circle} \mbox{$\mathbb{S}^1=\{z\in\cc:|z|=1\}$}.  It follows that 
{\em $\mathbb{S}^1$ is a \mbox{Warga} derivate container of
$f$ at $0$.}  On the other hand, {\em $\mathbb{S}^1$ is not
a GDQ of $f$ at $0$.}  We will actually show that
{\em if $\Lambda$ is any GDQ of $f$ at $0$, then
$0\in\Lambda$}. To see this, assume that
$0\notin\Lambda$, and pick a compact 
\mbox{neighborhood} $\Lambda'$ 
of $\Lambda$ such that $0\notin\Lambda'$. Observe 
that, if $x>0$, then we can make the change of variables
$t=1/s$ and conclude that 
\mbox{$f(x)=\int_{1/x}^\infty\frac{e^{is}}{s^2}\,ds$}.
Integration by parts, with $\xi=1/x$,  then yields
$f(x)=-i\xi^{-2}e^{i\xi}+
2i\int_{\xi}^\infty\frac{e^{is}}{s^3}\,ds$.
A \mbox{second} integration by parts then gives
$$
f(x)=-i\xi^{-2}e^{i\xi}+
2\xi^{-3}e^{i\xi}-6\int_{\xi}^\infty\frac{e^{is}}{s^4}\,ds\,.
$$
But $6|\int_{\xi}^\infty\frac{e^{is}}{s^4}\,ds|\le
6\int_{\xi}^\infty s^{-4}\,ds=2\xi^{-3}$.  Therefore
$|f(x)|=O(\xi^{-2})=O(x^2)$ as $x\downarrow 0$. Now,
since  $\Lambda$ is a GDQ of $f$ at $0$, $f$ must have
a factorization $f(x)=g(x)x$ such that $g$ is continuous
and $\Lambda'$-valued on some punctured neighborhood
$]-\alpha,\alpha[\backslash\{0\}$ of $0$.  Hence there exists
a $\delta\in\rr$ such that $\delta>0$ and
$|g(x)|\ge\delta$ \mbox{whenever} $0<x\le\alpha$. On the other hand,
the bound $|f(x)|=O(x^2)$ implies that
$|g(x)|=O(x)$ as $x\downarrow 0$, and we have reached a 
contradiction.

\ssection{Conclusion}\label{secconclu}
\noindent The remarks at the end of the previous 
section show that the two ``most general'' GDTs 
described so far---that is, GDQ and WMD---do not contain each other. 
In the \mbox{second} paper of this series we introduce a new GDT---the 
``\pigd''---that \mbox{contains} both.


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